Short answer: no.
Long answer: While the Greek lacks the definite article on theos in the clause under discussion, that doesn't mean the English should be translated with an indefinite article. Greek and English do not enjoy a one-to-one relationship between their words. There are times in Greek when the article is present but not translated into English. Likewise, there are places where the article is not present in the Greek but the English requires it, or in this case, requires something to show the definiteness of the word.
Example 1: John 18:16 in Greek literally says: "...the disciple, the other, the one known to the high priest..." That's horrible English. So it gets translated (rightly) as "the other disciple, who was known to the high priest." As you can see the word order changed coming into English as well as two definite articles dropping out.
Example 2: John 1:1 contains another example of a time without an article in Greek but needed in English. It says, "en arche 'en o logos..." that is (literally) "In beginning was the Word." Notice that there is no definite article before arche. However, even the New World Translation puts the article there. That is how it should be. To leave it out would cause confusion in the English "In a beginning was the Word..." That implies that there were multiple beginnings to the universe, but that isn't what the Bible teaches. It's a difference in Greek and English. Likewise, the Septuagint of Genesis starts with en arche.
The reason the clause at the end of John 1:1 lacks the article deals with rules of Greek grammar. English uses word order to drive the meaning of a sentence. We almost always have subjects first, then verbs, then the objects (excepted Yoda speech is). Greek doesn't use word order to differentiate between types of nouns. They use word order for emphasis (Hebrew does the same thing). To tell the difference in the subject and the object (both of which are nouns), Greek uses case endings. They can then put the object of the verb at the beginning of the clause with the subject after the verb and still know what the sentence means. In English, "dog bites man" and "man bites dog" mean entirely different things.
However, in Greek, they would put case endings on the nouns and comprehend the same meaning even with the word order switched around. In the following example, I am using case endings here as an illustration. [s] means subject, and [o] means object. In Greek there is no difference between "dog[s] bites man[o]" and "man[o] bites dog[s]." They mean the exact same thing. This works with action verbs, linking verbs are different, but the action verbs show how the Greek usually works.
The clause in question (which uses a linking verb) literally reads kai theos 'en 'o logos (literally "and God was the Word" but you won't find it translated that way for good reason). Notice that the word order is switched around with "God" at the front of the clause. Because the verb is a linking verb, the subject and object use the same case ending, the nominative. With a linking verb, the part of the clause that would be the object often drops the article (even though it would use it otherwise), especially when it is in front of the verb (as here). When the object of a clause is a noun like this, it is called the "predicate nominative" and Colwell's Rule allows the translation to indicate the definiteness of the word even when the Greek lacks the article.
In English, we don't put "the" in front of God to show definiteness. We capitalize it. That's what Greek scholars recognize in this verse.
